## Questions and Answers

Simplify the expression: $(2\sqrt{3} + 3\sqrt{2})^2$.

48 + 36√6

Simplify the expression: $4^{\frac{2}{3}} × 4^{\frac{1}{3}}$.

8

Simplify the surd: $5\sqrt{75}$.

25\sqrt{3}

If $a = 2^{\frac{1}{2}}$, $b = 2^{\frac{1}{4}}$, and $c = 2^{\frac{3}{4}}$, find the value of $a^2b^4c^2$.

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Simplify the expression: $\frac{4^{\frac{2}{3}}}{4^{\frac{1}{3}}}$.

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## Study Notes

### Surds and Indices

### Surds

- A surd is a value that, when multiplied by itself, gives a whole number (e.g. √2, √3, etc.)
- Surds can be simplified by finding the largest perfect square that divides the number under the square root, and then taking that square root out (e.g. √12 = √(4 × 3) = 2√3)
- Rationalising a surd involves eliminating the surd from the denominator of a fraction by multiplying both numerator and denominator by the surd (e.g. 1/√2 = √2/2)
- Surds can be added and subtracted when they have the same base and index (e.g. 2√3 + 3√3 = 5√3)
- Surds can be multiplied and divided by using the property √a × √b = √(a × b) and √a ÷ √b = √(a ÷ b)
- When adding and subtracting surds with different bases, convert to the same base by finding a common multiple of the bases (e.g. √2 + √8 = √2 + √(4 × 2) = √2 + 2√2)

### Indices

- Indices, also known as powers or exponents, are used to express very large or very small numbers in a more compact form (e.g. 2^3 means 2 to the power of 3, or 2 × 2 × 2)
- The index laws state that a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m-n), and (a^m)^n = a^(m×n)
- Indices can be used to simplify fractions by using the index laws (e.g. 2^3/2^2 = 2^(3-2) = 2^1 = 2)
- When simplifying fractions with indices, the numerator and denominator must have the same base, and the indices can be subtracted (e.g. 2^5/2^3 = 2^(5-3) = 2^2 = 4)

Note: These study notes provide a concise summary of the key concepts and formulas related to surds and indices at the 10th grade level in Australia. They are meant to serve as a quick reference guide for students to study and review these topics.

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## Description

Test your skills on surds and indices at the 10th grade level. This quiz covers rationalisation, simplification, multiplication, division, addition and subtraction of surds, as well as index laws and simplification of fraction with indices.